A211380 Number of pairs of intersecting diagonals in the interior and exterior of a regular n-gon.

0, 1, 5, 15, 42, 94, 189, 340, 572, 903, 1365, 1981, 2790, 3820, 5117, 6714, 8664, 11005, 13797, 17083, 20930, 25386, 30525, 36400, 43092, 50659, 59189, 68745, 79422, 91288, 104445, 118966, 134960, 152505, 171717, 192679, 215514, 240310, 267197, 296268, 327660

Offset: 3

Author

Martin Renner, Feb 07 2013

Keywords

• nonn
• easy

Links

• Eric Weisstein, Regular Polygon Division by Diagonals (MathWorld).
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A176145, A211379.

Programs

Maple

a:=n->piecewise(n mod 2 = 0,1/8*n*(n^3-11*n^2+43*n-58),n mod 2 = 1,1/8*n*(n-3)*(n^2-8*n+19),0);

Mathematica

Drop[CoefficientList[Series[x^4(2x^5-3x^4-7x^3-x^2-2x-1)/((x-1)^5(x+1)^2),{x,0,50}],x],3] (* or *) LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,5,15,42,94,189},50] (* Harvey P. Dale, Dec 03 2022 *)

Python

def A211380(n): return n*(n*(n*(n-11)+43)-58+(n&1))>>3 # Chai Wah Wu, Nov 22 2023

Formula

a(n) = 1/8*n*(n^3-11*n^2+43*n-58) for n even;

a(n) = 1/8*n*(n-3)*(n^2-8*n+19) for n odd.

a(n) = A176145(n) - A211379(n).

G.f.: x^4*(2*x^5-3*x^4-7*x^3-x^2-2*x-1) / ((x-1)^5*(x+1)^2). [Colin Barker, Feb 14 2013]